I just projected a feature class originally in Web Mercator (WKID 3857) to WKID 6318 (ESRI calls this "GCS_NAD_1983_2011") using ESRI's projection tool, and the same feature class to WKID 6318 (OGR calls this "NAD83(2011)") using ogr2ogr. They are very different: if I put them side by side in ArcMap they are 38" off in some places. What's more, I added the GCS_NAD_1983_2011 shapefile (the ESRI-projected shapefile) to ArcMap first and when adding the NAD83(2011) shapefile (the ogr-projected shapefile) ESRI prompted to use a transformation since they are different datums.

This is unexpected: I expected WKID 6318 to be the same between the two (ESRI and ogr). Why would they be so different?

This is the full description of GCS_NAD_1983_2011/ESRI (looking at the 'XY Coordinate System' tab in the Properties window of ArcCatalog):
GCS_NAD_1983_2011
WKID: 6318 Authority: EPSG

The important part is the SRID (WKID). Both are identical. Even though the datums have different names their parameters would be the same... and a rose by any other name... is still a rose!
– Michael StimsonJul 20 '17 at 0:04

Thanks for the response Michael. I get that the SRIDs are the same, which is why I would expect the projected layers to be the same. But they're not: when I add both layers to ArcMap I expect they would exactly overlay each other (or nearly so, within centimeters at most), but that's not what's happening: they're a few feet off in places.
– crbJul 20 '17 at 0:20

Yes that is odd, I suspect it has more to do with the transformation that is applied. I'm not an expert in northern hemisphere projections but from my own experience (AGD66 to GDA94 NTv2) the Esri transformation is more accurate compared to the OSR default or approximate transformation; I must admit though I'm not sure how to specify a datum transformation file for OSR. It would be good if you could compare both outputs with a known to be accurate dataset to prove in your situation which one is more correct.
– Michael StimsonJul 20 '17 at 0:28

My guess is that on the OGR-side, there's a proj.4 string that includes a +TOWGS84 parameter...or not. Was a transformation used when you converted from 3857 to 6318 in ArcGIS? If the same tfm wasn't used in OGR--different output.
– mkennedyJul 20 '17 at 19:50